To the OP, I would suggest you reach out to police in every state you are intending to visit and PRINT OUT THEIR RESPONSES TO YOU. Seeking legal information on the internet is sketchy at best. None of these people are going to testify for you at a trial if you need. If you have an email from the State Police stating XYZ and then you get arrested in that state, you have a much better chance beating he charges if you can produce documentation that you reached out to state law enforcement and received information that you believed to be true.
That being said, I would only have the gun in the car during the day. Even then, the gun would be dis-assembled and nowhere near ammo. If a police officer ever asked, I would be 100% forthright with them. Whenever you get to a hotel, I would immediately transfer the gun to the safe in the hotel room so I didn't have it in the car if not necessary.